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Theorem ssrabdv 3086
Description: Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 31-Aug-2006.)
Hypotheses
Ref Expression
ssrabdv.1  |-  ( ph  ->  B  C_  A )
ssrabdv.2  |-  ( (
ph  /\  x  e.  B )  ->  ps )
Assertion
Ref Expression
ssrabdv  |-  ( ph  ->  B  C_  { x  e.  A  |  ps } )
Distinct variable groups:    x, A    x, B    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem ssrabdv
StepHypRef Expression
1 ssrabdv.1 . 2  |-  ( ph  ->  B  C_  A )
2 ssrabdv.2 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  ps )
32ralrimiva 2442 . 2  |-  ( ph  ->  A. x  e.  B  ps )
4 ssrab 3085 . 2  |-  ( B 
C_  { x  e.  A  |  ps }  <->  ( B  C_  A  /\  A. x  e.  B  ps ) )
51, 3, 4sylanbrc 408 1  |-  ( ph  ->  B  C_  { x  e.  A  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1436   A.wral 2355   {crab 2359    C_ wss 2986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rab 2364  df-in 2992  df-ss 2999
This theorem is referenced by: (None)
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